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arXiv:2407.10444 · APS Open Science

Superfluid transition temperature of bond bipolarons with Coulomb interaction

Chao Zhang

Published May 20, 2026• 0 citations
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Abstract

Using numerically exact diagrammatic Monte Carlo simulations in the two-electron (single-bipolaron) sector, we explore the impact of long-range Coulomb repulsion on the dilute-limit Berezinskii-Kosterlitz-Thouless transition temperature T c of bipolarons on a two-dimensional square lattice. We study the bond Su-Schrieffer-Heeger model, in which bond phonons modulate the electron hopping. In the absence of long-range repulsion, this model was shown to support small, light bipolarons with a comparatively high transition temperature [C. Zhang, J. Sous, D. R. Reichman, M. Berciu, A. J. Millis, N. V. Prokof’ev, and B. V. Svistunov, Bipolaronic high-temperature superconductivity, ]. Here, we find that long-range Coulomb repulsion suppresses the optimal T c but leaves it appreciable over a broad parameter window, including the adiabatic regime ω / t = 0.5 at a representative Coulomb strength V = U / 10 (with U the on-site repulsion). Our results provide controlled single-bipolaron inputs for dilute-limit T c estimates in the presence of long-range repulsion.

DOI
10.1103/zddz-992r

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Using numerically exact diagrammatic Monte Carlo simulations in the two-electron (single-bipolaron) sector, we explore the impact of long-range Coulomb repulsion on the dilute-limit Berezinskii-Kosterlitz-Thouless transition temperature T c of bipolarons on a two-dimensional square lattice. We study the bond Su-Schrieffer-Heeger model, in which bond phonons modulate the electron hopping. In the absence of long-range repulsion, this model was shown to support small, light bipolarons with a comparatively high transition temperature [C. Zhang, J. Sous, D. R. Reichman, M. Berciu, A. J. Millis, N. V. Prokof’ev, and B. V. Svistunov, Bipolaronic high-temperature superconductivity, ]. Here, we find that long-range Coulomb repulsion suppresses the optimal T c but leaves it appreciable over a broad parameter window, including the adiabatic regime ω / t = 0.5 at a representative Coulomb strength V = U / 10 (with U the on-site repulsion). Our results provide controlled single-bipolaron inputs for dilute-limit T c estimates in the presence of long-range repulsion.
We employ the diagrammatic Monte Carlo method with a lattice path-integral formulation for both electron and phonon degrees of freedom to investigate the formation and properties of bond polarons and bipolarons on a two-dimensional triangular lattice. In the deep adiabatic regime ($\ensuremath{\omega}/t<1.0$), single polarons remain light with a small effective mass, while bipolarons remain compact and lightweight, resulting in a high superfluid transition temperature Tc{T}_{c}Tc​. We systematically study the dependence of Tc{T}_{c}Tc​ on electron-phonon coupling strength and on-site interaction in the bond bipolaron model. Our results show that a moderate on-site repulsion enhances Tc{T}_{c}Tc​ by stabilizing compact yet lightweight bipolarons, leading to high-Tc{T}_{c}Tc​ superconductivity in the dilute limit. Notably, the triangular lattice sustains relatively high Tc{T}_{c}Tc​ across a wide range of phonon frequencies, outperforming square lattice geometry. This enhancement arises from the higher coordination number and the bond-centered nature of the electron-phonon coupling. These findings suggest that triangular lattice geometry offers a promising platform for realizing high-Tc{T}_{c}Tc​ bipolaronic superconductivity.
We analyze the transition temperature Tc{T}_{c}Tc​ of bipolaronic superconductivity in a bond Su-Schrieffer-Heeger (bond-SSH) model---also known as a bond Peierls model---where bond phonons modulate the electron hoppings. Using a semiclassical instanton approximation justifiable in the adiabatic limit of slow phonons, we find that the bipolaron mass is only weakly enhanced in contrast to the typically large mass enhancement found in models where the phonons couple to electron density. Strikingly, in the ``atomic'' limit, we find a bond bipolaron can freely slide within a continuously degenerate energy manifold, explaining the origin of the weak mass enhancement in the strong coupling limit. A gas of these bipolarons can undergo a superfluid transition at a critical temperature for which we obtain an upper bound. This bound is exponentially larger than that in the Holstein model. Our study provides an analytical understanding of the mechanism behind the high-Tc{T}_{c}Tc​ bipolaronic superconductivity numerically observed in [Phys. Rev. X 13, 011010 (2023)].
We present a diagrammatic Monte Carlo study of the properties of the Hubbard-Holstein bipolaron on a two-dimensional square lattice. With a small Coulomb repulsion UUU and with increasing electron-phonon interaction, and when reaching a value about two times smaller than the one corresponding to the transition of light polaron to heavy polaron, the system suffers a sharp transition from a state formed by two weakly bound light polarons to a heavy, strongly bound on-site bipolaron. Aside from this rather conventional bipolaron a new bipolaron state is found for large UUU at intermediate and large electron-phonon coupling, corresponding to two polarons bound on nearest-neighbor sites. We discuss both the properties of the different bipolaron states and the transition from one state to another. We present a phase diagram in parameter space defined by the electron-phonon coupling and UUU. Our numerical method does not use any artificial approximation and can be easily modified to other bipolaron models with longer range electron-phonon and∕or electron-electron interaction.
Polarons originating from phonon displacement modulated hopping have relatively light masses and thus are of significant current interest as candidates for the bipolaron mechanism of high-temperature superconductivity [Sous et al., Phys. Rev. Lett. 121, 247001 (2018)]. We observe that the bond model, when the dominant coupling comes from atomic vibrations on lattice bonds, can be solved by efficient sign-free Monte Carlo methods based on the path-integral formulation of the particle sector in combination with either the (real-space) diagrammatic or Fock-path-integral representation of the phonon sector. We introduce the corresponding algorithms and provide illustrative results for singlet bipolarons in two dimensions. The data suggest that the route towards high-temperature superconductivity (if any) in the multiparametric space of the model lies between the Scylla of large size of moderately light bipolarons and the Charybdis of large mass of compact bipolarons. As a result, on-site repulsion is helping sss-wave superconductivity in sharp contrast with existing expectations.
Polarons with different types of electron-phonon coupling have fundamentally different properties. When the dominant interaction is between the electron density and lattice displacement, the momentum of the ground state does not change and the polaron gets exponentially heavy at strong coupling. In contrast, one-dimensional Peierls/Su-Schrieffer-Heeger (PSSH) polarons with interaction originating from displacement-modulated hopping feature a shift of the ground-state momentum to finite values and moderate values of effective mass as coupling is increased [D. J. J. Marchand et al., Phys. Rev. Lett. 105, 266605 (2010)]. Based on Diagrammatic Monte Carlo method, we investigate whether unusual properties of PSSH polarons depend on the type of the displacement-modulated hopping and to what degree they survive in higher dimension. We study two different PSSH models: with bosonic degrees of freedom residing on sites (model A) and bonds (model B) of the two-dimensional square lattice. For model A, we find that in both adiabatic and intermediate regimes, the momentum of the ground state experiences a continuous transition from zero to a finite value as a function of coupling strength. The transition is driven by quadratic instability of the dispersion function, implying that effective mass diverges at the critical point, and then decreases in an anisotropic fashion with increasing coupling. Unexpectedly, for model B, the momentum of the ground state always stays at zero and the effective mass increases monotonously with coupling. The increase is far from exponential and tends to level off at strong interaction, resulting in relatively light polarons. Having light polarons in the strong coupling regime is crucial for the bipolaron mechanism of high-temperature superconductivity [J. Sous, M. Chakraborty, R. V. Krems, and M. Berciu, Phys. Rev. Lett. 121, 247001 (2018).]
We study a lattice bipolaron on a staggered triangular ladder and triangular and hexagonal lattices with both long-range electron-phonon interaction and strong Coulomb repulsion using a novel continuous-time quantum Monte Carlo algorithm to solve the two-particle Coulomb-Fröhlich model. The algorithm is preceded by an exact integration over phonon degrees of freedom, and as such is extremely efficient. The bipolaron effective mass and radius are computed. Bipolarons on lattices constructed from triangular plaquettes have a novel crablike motion, and are small but very light over a wide range of parameters. We discuss the conditions under which such particles may form a Bose-Einstein condensate with high transition temperature, proposing a route to room temperature superconductivity.